How do I fix this issue with my model's texture
The reason parts of the mesh appear black, is because two vertices are in the same location, and blender renders them as black.

Deleting the vertices:
Select the vertices you don't need, and press Delete, and select Dissolve Vertices. This will remove the vertices and remove the black overlapping mesh int he Rendered view.

EDIT:

Merging the vertices
Instead of deleting them, it may be better to merge them. Select two vertices in a similar location, and press Alt+M and select At Center, if you want to merge the vertices in the middle of both vertices. This will make them into one vertical, without removing any other lines the deleting method would.